"If and only if" is a bidirectional implication. We have to prove both directions: 1. solvable by radicals implies the splitting field has a solvable Galois group; 2. the splitting field having a solvable Galois group implies solvability by radicals.
-
Show this thread
-
But once we do that we set up the coup de grace: now we only need to compute the Galois group for a quintic polynomial (not so hard) and then show it is not solvable. This is probably not easy to follow if you don't know these terms. That's ok.
1 reply 0 retweets 8 likesShow this thread -
What's elegant about this is it avoids a tremendous amount of tedium by doing a lot of overhead work first. We establish that the problem we care about looks like this other thing. Once we do that, the rest falls like dominoes.
2 replies 0 retweets 11 likesShow this thread -
A good proof is like a good chef cooking a complex meal. It's chaos and complexity and drama! And then, all of a sudden, everything comes together at the end. It can be hard to understand what they're doing at first... but them BAM.
2 replies 1 retweet 12 likesShow this thread -
Have you ever seen that clip of Susan Boyle at Britain's Got Talent? It's a trainwreck at first. She's dowdy, and old. She's out of her element. "I Dreamed a Dream?". Simon's making fun of her. She is mumbling her words... And then she sings.https://www.youtube.com/watch?v=RxPZh4AnWyk …
1 reply 3 retweets 17 likesShow this thread -
That is what a perfect proof feels like.
3 replies 0 retweets 10 likesShow this thread -
Becoming good at math means being able to identify the setup work that needs to take place. This is not a special skill. Much like any professional, if you become good at something you know how to get started, whether it's starting a software project or a new patient record or
1 reply 0 retweets 10 likesShow this thread -
But great mathematicians can see multiple steps into the future in this process, to intuit connections that are not obvious. Throughout my math undergrad, I viewed math as a mostly mechanical, procedural process. Which made sense for my discipline (scientific computing).
1 reply 0 retweets 9 likesShow this thread -
But as I shifted more into formal analysis I began to finally understand proof. It took a mental maturity well-beyond my undergrad years to get there. It really took me until my late-20s. And I'm still learning that process.
2 replies 0 retweets 12 likesShow this thread -
Replying to @EmilyGorcenski
Some top mathemeticians do ground-breaking work early, though, yes? (Not said dismissively of your own learning process.) I was always jealous of my math PhD colleagues, whose dissertations were often short papers where they worked out theorems. So different than humanities!
2 replies 0 retweets 0 likes
This is a common myth that is largely a product of western academic structures. But it is true that math PhDs can be (much) shorter.
Loading seems to be taking a while.
Twitter may be over capacity or experiencing a momentary hiccup. Try again or visit Twitter Status for more information.